Kremmit raises an excellent point: most microswitch controls have a normally-closed terminal in addition to the normally-open one. Connect the "common" terminal to ground, (as you probably will with most encoders) and "N.O." to the encoder input, and then you can use the third, "Normally Closed" terminal to create your LED circuit.
You can make the LED circuit completely passive - no gates or transistors or anything. Just connect the cathode side of each LED to ground, and to the NC terminal connect the anode side of the LED and also a resistor going to +5V. This is what will happen:
When the switch is not pressed, the "NC" terminal will be connected to ground. The pull-up resistor on that line will pass current straight to ground. Because both sides of the LED will be at zero volts, it will not light.
When the switch is pressed, the "NC" terminal will no longer have a direct connection to ground, so the current flowing from +5V through the pull-up resistor will flow through the LED, lighting it.
Of course, the +5V source may have to come from your encoder board for this to work... The +5V has to be +5V relative to the ground node, and since you're connecting the switch's common terminal to the encoder's ground line... well, there you go.
You needn't wait to have a physics class to learn this stuff - you probably have access to teachers and textbooks which can help you with this. You can also look for basic electronics tutorials on-line - though I'm a bit wary of that approach because there's so much bad information out there.
Here are some basics.
First of all, you can think of the entire circuit as a collection of "nodes". Any part of a device that has a lead sticking out, that's a circuit node. When two nodes are connected together with a wire, they're basically the same node. (Certainly for our purposes that's a fine assumption.) At any given time, a particular node will have a particular voltage level relative to the circuit's "ground" node.
Voltages are always measured relatively between two nodes. It's meaningless to say that a node has a particular voltage without making some kind of statement about what the measurement is taken relative to. Commonly it'll be taken relative to ground, but it's also helpful sometimes to measure two nodes relative to each other. This is useful when you're talking about two nodes on the same device: for instance, the voltage across the two nodes of a resistor can tell you how much current is passing through the resistor. (The greater the voltage difference, the greater the current...)
Current represents the flow-rate of electrons in the circuit. Current flows through wires, and between terminals of multi-terminal devices. The basic rule of current is called Kirchoff's Current Law, which states that the amount of current flowing into a node must equal the amount of current flowing out of a node.
Building simple circuits is sometimes a matter of balancing simple math problems, making sure the amount of current flowing through devices is great enough to operate them (when desired) but not so great that they burn out. Different devices have different relationships that determine how much current will pass based on how much voltage is across them. Resistors and light bulbs and such are generally linear, meaning that if you double the voltage drop, you roughly double the current. LEDs are different, however. They have a particular voltage operating range. If you exceed that, the current increases greatly. Double an LED's voltage and you'll likely burn it out rather quickly.
IMO understanding the basics of nodes, voltages, and currents is critically important in building circuits and understanding how they work. You can muddle through without it, lots of people do, but you'll fare better if you know the mechanics of the devices you're using.
I'm doing similar work for my cabinet, and this thread has raised some good points for me, too. In particular, the need for the LED driver's ground to match the game encoder's ground in order for the LED driver and game encoder to monitor the same signal from the controls was something I hadn't thought of. (If I have two USB devices doing the two jobs, the two grounds should be the same... probably both being connected to the shielding and, eventually, indirectly, to the computer's casing and the AC line ground... So maybe my design is OK... But what happens if I want to drive some stuff from a separate AC-DC adaptor? I figure it's best to give that one some extra thought.)